Ants on a square, and how many collisions to expect

Question

Four ants sit at the four corners of a square. Simultaneously each ant picks one of the two edges at its corner uniformly at random and walks along it at the same speed.

What is the probability that no two ants collide? And what is the expected number of head-on (edge) collisions?

Accepted Answer

No collision. As with any polygon, the ants avoid every collision only if they all walk the same way around: all four clockwise, or all four counterclockwise. In any other pattern, some adjacent pair walks toward each other on a shared edge or into a shared corner. Each ant picks between 2 directions independently, so there are 2^4 = 16 equally likely configurations, of which exactly 2 are collision-free: P(	ext{no collision}) = \frac{2}{16} = \frac{1}{8}. Expected head-on collisions. Count an edge as a head-on collision if both its endpoint ants step onto it (walking toward each other). The…

Ants on a square, and how many collisions to expect

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